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Semiconductors / 半導体(Physics of semiconductors)
Institute for Solid State Physics, University of Tokyo
Shingo Katsumoto
Lecture on
2021.4.14 Lecture 02
Review of last week
2
Crystal structure • Basis, primitive cell, unit cell
• Lattice, Bravais lattice
• Reciprocal lattice
• Brillouin zone
Semiconductor materials • Group IV: C, Si, Ge, Sn, SiC, SixGe1-x
• Group III-V: GaAs, InP, AlAs, InAs, GaSb, …
• III-N: GaN, InN, …
• Group II-VI: CdTe, HgTe, …
Crystal growth
Bulk Thin film
• Czochralski
• Bridgman
• Floating zone
• MOCVD
• MBE
Chapter 1 Crystal structure and crystal growth
Chapter 2 Energy bands, effective mass approximation
Nearly free electron model, empty lattice approximation
Tight-binding approximation (TBA)
3
Single atom on single unit cell
Single atom Hamiltonian:
𝑇: kinetic energy, 𝑢: atomic potential
𝜙𝑛: eigenfunctions for 𝑅𝑖 = 0
Lattice periodic function
: Bloch form
Tight binding approximation (2)
4Hopping integral
Crystal field contribution
Tight binding approximation (3)
5
𝑡𝑛 nearest neighbor only = 𝑡
Cosine band with the width of 4𝑡.
Methods for obtaining band structure
6
Experiments • Hot electron transport
• Optical absorption
• Electroreflectance
• Cyclotron resonance
• Photoemission spectroscopy
Empirical calculation • Pseudo-potential approximation
• 𝑘 ∙ 𝑝 perturbation
ab-initio calculation • Local density approximation
• Augmented plane wave
• Generalized gradient approximation
Angle resolved photoemission spectroscopy (ARPES)
7see, e.g. for short review B. Lv, T. Qian, H. Ding, Nature Reviews Physics 1, 609 (2019).
T. C. Chang et al.
Phys. Rev. B21,
3513 (1980).
GaAs
Angle resolved photoemission spectroscopy (ARPES) (2)
8
Soft-X ray ARPES of GaAs
Strocov et al., J. Electron Spectroscopy and Related Phenomena 236, 1 (2019).
Bands below Fermi energy can be detected
Plane wave expansion
9
Crystal Schrodinger equation:
Bloch function (omit band index)
Fourier expansion
(∵ lattice periodicity)
(1)
(2)
(3)
(2), (3) → (1) (4)
(5)Each term in the sum over G is
zero in (4)
For (5) to have non-trivial solution
Pseudo-potential calculation method
10
→ We need 𝑉𝑮
Pseudo potential method: 1. Only consider valence bands and conduction bands around the
Fermi level. Effect of core electrons in renormalized into
periodic potential.
2. Replace real potential with pseudo potential which gives
similar tailing of wavefunction.
V(r)
uk(r)ukp(r)
Vp(r)
band structure: almost determined in skirt characteristics
Pseudo potential calculation method (2)
11
simplest example
𝑉 𝑟 = −𝑍𝑒
𝑟Replacement with
pseudo potential
Crystal pseudo
potential
Fourier transform:
𝑁: unit cell number
Ω: unit cell volume
: vectors pointing nuclei in the unit cell
: form factor (Fourier transform of 𝑊𝑝(𝑟)) depends only on potential form
: structure factor depends only on internal structure of unit cell
Empirical pseudo potential calculation for fcc semiconductors
12
ex) GaAs
distance Points number
0 (0,0,0) 1
3 (1,1,1),… 8
2 (2,0,0),… 6
8 (2,0,2),… 12
11 (3,1,1),… 24
Form factors obtained from fitting to optical experimentDistance from the origin and number
of points in reciprocal lattice
Band structure of Si and GaAs
13
100
010
001
conduction valley energy surface
Si
GaAsspin-orbit interaction is not taken into account
Definition of effective mass
14
Group velocity of wavefunction with
energy eigenvalue 𝐸𝑛(𝑘)
Acceleration
With definition
: inverse effective mass tensor :effective mass tensor
Energy surface measurement (cyclotron resonance)
15
Motion of charged particle in magnetic field:
cyclotron motion in the plane perpendicular
to the magnetic field Landau quantization
Cyclotron frequency
Optical pumping → microwave absorption
𝜔𝑐 → cyclotron mass 𝑚c
Energy surface: ellipsoid
𝜃B
Electron-hole distinction ← circular polarization
Dresselhaus, Kip, Kittel, Phys. Rev. 98, 368 (1955).
Ge
(24 GHz microwave)
Cyclotron resonance
16
k·p perturbation
17
Crystal Schrodinger equation:
Bloch function
(1)
(2)
Equation for lattice periodic
function
Perturbation by 𝒌-dependent term
Good approximation for small k → band edge information
(a) In the case of no degeneracy
(3)
k·p approximation (2)
18
(b) In the case of n-fold degeneracy in 𝑢00(𝒓)
Approximate the perturbed wavefunction
Substitute to the equation for u
Taking inner product with |0𝑖
(4)
For eq.(4) to have non-trivial solution
orthogonal
Spin-orbit interaction
19
Spin-orbit Hamiltonian
spin-operator
From identity
eigenequation
G-band edges of diamond and zinc-blende semiconductors
20
Ec
Ev1
Ev2
Ev3
electron
heavy hole
light hole
spin-split off
k
Eunperturbed equation:
Diamond, zinc-blende: formed from 𝑠𝑝3 orbitals
perturbation
non-zero element
𝛼: spin coordinate
G-band edges of diamond and zinc-blende semiconductors (2)
21
G-band edges of diamond and zinc-blende semiconductors (3)
22
Ec
Ev1
Ev2
Ev3
electron
heavy hole
light hole
spin-split off
k
EEigenvalue equation
Ignoring the term 𝑃 2𝑘2 we finally obtain:
Band warping, a conventional way to get band parameters
23
Ev1
Ev2
heavy hole
light hole
k
E
constant-energy
surface
Summary of 𝑘 ∙ 𝑝 second order perturbation
Table of band parameters
24Lundstrom “Fundamentals of Carrier Transport” (Cambridge, 2000).
Graphene: A two-dimensional material (another example of TBA)
van der Waals
covalent
Graphite Graphene
Graphene lattice/reciprocal lattice structure
Atomic orbitals Honeycomb lattice
Graphene lattice/reciprocal lattice structure
Lattice: unit cell Reciprocal lattice
Tight binding model
Eigenvalues:
Linear combination
Sublattice wavefunction
tight-binding
tight-binding
Hamiltonian equation
Sublattice transition term
Take the nearest neighbor approximation:
Dirac points in k -space
A Dirac point
𝑘𝑥
𝑘𝑦